# Sum of Arithmetic Progression/Examples/a0 = i, d = 2+2i

## Example of Sum of Arithmetic Progression

Let $A_n$ be the arithmetic progression of $n$ terms defined as:

 $\displaystyle A_n$ $=$ $\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {a_0 + \paren {2 + 2 i} k}$ $\displaystyle$ $=$ $\displaystyle i + \paren {2 + 3 i} + \paren {4 + 5 i} + \paren {6 + 7 i} + \dotsb + \paren {2 n - 2 + \paren {2 n - 1} i}$

Then:

$A_n = n \paren {n - 1} + n^2 i$

## Proof

 $\displaystyle A_n$ $=$ $\displaystyle n \paren {i + \frac {n - 1} 2 \paren {2 + 2 i} }$ Sum of Arithmetic Progression: $a_0 = i$, $d = 2 + 2 i$ $\displaystyle$ $=$ $\displaystyle n \paren {i + \paren {n - 1} \paren {1 + i} }$ $\displaystyle$ $=$ $\displaystyle n \paren {i + n - 1 + n i - i}$ $\displaystyle$ $=$ $\displaystyle n \paren {n - 1 + n i}$ $\displaystyle$ $=$ $\displaystyle n \paren {n - 1} + n^2 i$

$\blacksquare$